Download Electromagnetic interaction of a conducting cylinder with a magnetic

PAMM · Proc. Appl. Math. Mech. 12, 579 – 580 (2012) / DOI 10.1002/pamm.201210278
Electromagnetic interaction of a conducting cylinder with a magnetic
dipole caused by steady rotation
Sonja Engert1,2,∗ , Thomas Boeck1 , and André Thess1
1
2
Dept. of Electrothermal Energy Conversion, Ilmenau University of Technology, P.O. Box 100565, Ilmenau, Germany
Dept. of Thermodynamics and Magnetofluiddynamics, Ilmenau University of Technology, P.O. Box 100565, Ilmenau,
Germany
The motion of a conductor in a magnetic field induces eddy currents whose interaction with the field produces Lorentz
forces opposing the motion. One can determine the velocity of the conductor from the force on the magnet system since
the latter is equal but opposite to the Lorentz force on the conductor. This contactless method is known as Lorentz force
velocimetry (LFV). We study an idealized configuration of LFV, i.e. a rotating solid cylinder interacting with a point dipole.
The understanding of parameter influences in this setup can be helpful for more realistic configurations. We use a purely
kinematic approach appropriate for low magnetic Reynolds numbers. Numerical results for small and large distances between
dipole and cylinder have been obtained with the commercial software COMSOL Multiphysics.
c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Problem formulation
We consider an infinitely long, electrically conducting, rotating cylinder with conductivity σ, radius R and velocity ~v =
(0, −Ωz, Ωy)T , where Ω is the angular velocity, under the influence of the inhomogeneous magnetic field of a point dipole.
~ = (µ0 /4π)(3~r · m
The dipole is located above the cylinder at a distance h. The magnetic field of the dipole is B
~ ~r/r5 − m/r
~ 3 ),
where ~r denotes the position in a shifted coordinate system with the dipole at the origin. The vector m
~ = (mx , my , mz )T is
the magnetic moment. The geometry is illustrated in figure 1.
Fig. 1: Rotating cylinder near a magnetic point
dipole.
Fig. 2: Computational grid used in the COMSOL simulations: y-z-view (left),
x-y-view (right).
The movement induces eddy currents in the cylinder. Assuming that the magnetic Reynolds number Rm = µ0 σΩR2 is
~ and the condition ∇ · ~j = 0. This
small, the eddy current density is determined by Ohm’s law ~j = σ(−∇φ + ~v × B)
~ as applied magnetic field, is known as quasistatic approximation. The
formulation based on the electric potential φ and B
induced currents are confined to the conductor, i.e. the normal
component of ~j vanishes on the surface. TheR Lorentz force
R
~
~
~ . The torque on the cylinder is T~ =
~
on the cylinder is the integrated density fL , i.e. FL = cyl j × BdV
~r × f~L dV .
cyl
The problem is made dimensionless using the characteristic length R, the characteristic velocity ΩR and the characteristic
magnetic induction µR0 m
3 . The nondimensional variables are
jnew =
2
j
F
T
h
, Fnew =
, Tnew =
, hnew = .
2
2
3
2
3
2
2
σΩRµ0 mR
σΩRµ0 m R
σΩRµ0 m R
R
Numerical simulations
We solve the nondimensional problem ∇2 φ = 2Bx resulting from Ohm’s law and ∇·~j = 0 with the FEM software COMSOL.
A typical FEM mesh is shown in figure 2. Starting from a homogeneous grid in the middle of the cylinder the round structure
was discretized by angular elements, which become thinner near the boundaries. Moreover, the mesh is very fine directly
under the dipole. Depending on the distance h the mesh could be adjusted in several properties, like the cylinder length, the
number of grid elements, the ratio of the element size in the center and on the boundary (in x-direction). The cylinder length
∗
Corresponding author: e-mail [email protected], phone +49 3677 69 1187
c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
580
Section 13: Flow control
a)
b)
Fig. 3: Nondimensional Lorentz force (a) and torque (b) on a rotating cylinder for a wide range of dipole distances. The torque vanishes
when the magnetic moment points along the cylinder axis.
a)
b)
Fig. 4: Comparison of simulation results and analytical calculation of the Lorentz force (a) and torque (b) for small dipole distances.
of the simulation model was adapted in order to approximate the force in the infinite case with 95% accuracy. The results for
Lorentz force and torque for the three main orientations of the dipole are illustrated in figure 3.
3
Analytical approximations
For very small dipole distances the problem becomes equivalent to that of a thick translating infinite plate with velocity
v = ΩR in the y-direction. The Lorentz force and torque in this case can be calculated analytically [1,2]. For a vertical dipole
the results for the plate are
Fy0 =
1 µ20 m2 σv
1 µ20 m2 σv
,
T
=
−
.
x0
128π
h3
128π
h2
(1)
Figure 4 shows that the simulation results are in very good agreement with eq. (1) for dipole distances h < 0.1.
In case of large h the dependencies on the distance change because of the finite width of the cylinder. In Figure 3 the
force decays as h−6 and the torque as h−5 . In analogy with the translating conducting square bar studied in [3], we have
formulated and solved an analytical approximation for this case by asymptotic analysis in the small parameter = R/h. It
allows us to exploit the slow variation of the induced currents along the cylinder axis by scaling the coordinate x with . A
regular perturbation expansion leads to two-dimensional Poisson problems for the electric potential at the different orders of
approximation. Force and torque are zero for the leading order of approximation 0 . In the next order of approximation 1 the
force is non-zero and the result is in good agreement with simulation results for dipole distances h > 100. In this way we
have confirmed the force decay rate h−6 and the fact that FL has its maximum for the vertical dipole orientation. The details
of this analysis and further numerical results will be published elsewhere.
Acknowledgements We are grateful to DFG for financial support in the framework of Research Training Group “Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing” (grant GRK 1567/1).
References
[1] A. Thess, E. Votyakov, B. Knaepen, and O. Zikanov, New J. Phys 9, 299 (2007).
[2] J. Priede, D. Buchenau, and G. Gerbeth, J. Appl. Phys. 110, 034512 (2011).
[3] M. Kirpo, S. Tympel, T. Boeck, D. Krasnov, and A. Thess, J. Appl. Phys. 109, 113921 (2011).
c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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